Types of Triangles

• A triangle with three congruent sides is called an equilateral triangle. The measures of the three
interior angles of such a triangle are also equal, and each measure is 60°.

• A triangle with at least two congruent sides is called an isosceles triangle. If a triangle has two
congruent sides, then the angles opposite the two sides are congruent. The converse is also true i.e. if the angles opposite the two sides are congruent then the sides are congruent.

• A triangle with an interior right angle is called a right triangle. The side opposite the right angle
is called the hypotenuse; the other two sides are called legs.

Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares
of the lengths of the legs.

If we take the length of the hypotenuse to be c and the length of the legs to be a and b then this theorem tells us that:

c2 =a2 + b2

The Pythagorean Theorem can be used when we know the length of two sides of a right triangle and we need to get the length of the third side.

Example: Find the length of one side of a right triangle if the
length of the hypotenuse is 10 inches and the length of the other
side is 9 inches.

Special Right Triangles

One special right triangle is an isosceles right triangle which has two congruent sides and angles 45°-
45°- 90°.

Applying the Pythagorean theorem to such a triangle shows that the lengths of its sides are in the ratio of .

The other special right triangle is a 30°-
60°- 90° right triangle, which is half of an equilateral triangle, as
indicated below. The lengths of the sides of a 30°-
60°- 90° triangle are in the ratio of .

Area of a Triangle

The area of a triangle equals one-half the product of the length of a base and the height corresponding
to the base.

Area =

In the figure below, the base is denoted by b and the corresponding height is denoted by h. The height of a triangle is the perpendicular distance from a vertex to the base of the triangle.

Any side of a triangle can be used as a base; the height that corresponds to the base is the perpendicular line segment from the opposite vertex to the base (or to an extension of the base).

Congruent Triangles

Two triangles that have the same shape and size are called congruent triangles. More precisely, two
triangles are congruent if their vertices can be matched up so that the corresponding angles and the
corresponding sides are congruent.

The following three propositions can be used to determine whether two triangles are congruent by
comparing only some of their sides and angles.
• If the three sides of one triangle are congruent to the three sides of another triangle, then the
triangles are congruent.
• If two sides and the included angle of one triangle are congruent to two sides and the included
angle of another triangle, then the triangles are congruent.
• If two angles and the included side of one triangle are congruent to two angles and the included
side of another triangle, then the triangles are congruent.

Similar Triangles

Two triangles that have the same shape but not necessarily the same size are called similar triangles.
More precisely, two triangles are similar if their vertices can be matched up so that the corresponding
angles are congruent or, equivalently, the lengths of corresponding sides have the same ratio, called the
scale factor of similarity.

When we say that triangles ABC and DEF are similar, it is assumed that angles A and D are congruent,
angles B and E are congruent, and angles C and F are congruent, as shown in the figure below. In other
words, the order of the letters indicates the correspondences.

If the above two triangles are similar
then

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